19.1 Basic Vectorization

To a very good first approximation, the goal in vectorization is to
write code that avoids loops and uses whole-array operations. As a
trivial example, consider

for i = 1:n
for j = 1:m
c(i,j) = a(i,j) + b(i,j);
endfor
endfor

compared to the much simpler

c = a + b;

This isn’t merely easier to write; it is also internally much easier to
optimize. Octave delegates this operation to an underlying
implementation which, among other optimizations, may use special vector
hardware instructions or could conceivably even perform the additions in
parallel. In general, if the code is vectorized, the underlying
implementation has more freedom about the assumptions it can make
in order to achieve faster execution.

This is especially important for loops with "cheap" bodies. Often it
suffices to vectorize just the innermost loop to get acceptable
performance. A general rule of thumb is that the "order" of the
vectorized body should be greater or equal to the "order" of the
enclosing loop.

As a less trivial example, instead of

for i = 1:n-1
a(i) = b(i+1) - b(i);
endfor

write

a = b(2:n) - b(1:n-1);

This shows an important general concept about using arrays for indexing
instead of looping over an index variable. See Index Expressions.
Also use boolean indexing generously. If a condition needs to be tested,
this condition can also be written as a boolean index. For instance,
instead of

for i = 1:n
if (a(i) > 5)
a(i) -= 20
endif
endfor

write

a(a>5) -= 20;

which exploits the fact that a > 5 produces a boolean index.

Use elementwise vector operators whenever possible to avoid looping
(operators like .* and .^). See Arithmetic Ops. For
simple inline functions, the vectorize function can do this
automatically.

: vectorize(fun)

Create a vectorized version of the inline function fun by replacing
all occurrences of *, /, etc., with .*, ./, etc.

This may be useful, for example, when using inline functions with numerical
integration or optimization where a vector-valued function is expected.

Also exploit broadcasting in these elementwise operators both to avoid
looping and unnecessary intermediate memory allocations.
See Broadcasting.

Use built-in and library functions if possible. Built-in and compiled
functions are very fast. Even with an m-file library function, chances
are good that it is already optimized, or will be optimized more in a
future release.

For instance, even better than

a = b(2:n) - b(1:n-1);

is

a = diff (b);

Most Octave functions are written with vector and array arguments in
mind. If you find yourself writing a loop with a very simple operation,
chances are that such a function already exists. The following functions
occur frequently in vectorized code: